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ON THE FEYNMAN PATH INTO THE SUN
by
Yung-Ching Liang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
August 2013
Copyright 2013 Yung-Ching Liang

This study deals with solar-physics applications of a recent equation-of-state formalism based on the formulation of the so-called ""Feynman-Kac (FK) representation"". This formalism leads to an exact virial expansion of the thermodynamic functions in powers of the particle densities of a Coulomb plasma (""exact"" here refers to the accuracy of the low-order virial coefficients of the expansion). By taking advantage of the exact and analytic form of this virial expansion, we can probe the thermodynamic properties of the solar interior, both in detail and to an accuracy that has so far not been achieved with currently available equation-of-state formalisms. For reacting plasmas, virial-expansion equation of state have an intrinsic problem when the plasma is less than fully ionized. Fortunately, in most parts of the Sun's interior, the plasma is almost completely ionized. Therefore, the FK virial equation of state can be applicable, but only in the deeper (and hotter) solar interior. The precise boundary of the domain of validity of the virial expansion depends on the elements included in the formalism. Since the computational effort increases tremendously with an increasing number of chemical elements, here, we choose to represent the heavier elements by a single one, oxygen. This approximation is reasonable for the following reason. In the Sun, the major heavy elements (that is, the elements other than H and He which together comprise more than 98% of the mass fraction) are C, N and O. They all have a similar effect in the equation of state. Then, the other elements, such as Fe and Ne, are so little abundant that they have altogether a very small effect in the equation of state (although they are very important in spectroscopy). Assuming the relevant constituents of the Sun to be H, He, and O, we have examined the effects on thermodynamic quantities from each of these components. With the aid of the FK formalism, we have studied the influences of the contributions to the respective different partial pressure. This not only helps us to understand the intrinsic challenge of applying the virial expansion, but also to analyze the various effects quantitatively. We have also tackled the aforementioned main limitation of the virial expansion, that is, its breakdown for relatively low temperatures, where the plasma is clearly less than fully ionized. For this, we have replaced the virial equation for these outer regions for the Sun by another, similar equation of state, the so-called ""scaled low temperature (SLT)"" expansion. Current versions of SLT are so far only for one chemical element, therefore restricted to the hydrogen part of the solar plasma. However, since hydrogen is the most abundant element in the Sun (more than 90% by number), SLT in its current form is already useful, if one adds the contributions of the remaining elements (He and heavy elements) using conventional equation-of-state formalisms. In such a procedure, an improved H part and a conventional treatment of the other elements will none the less lead to a net improvement of the overall equation of state compared to entirely conventional formalisms. Our results show (i) that the SLT EOS is not only consistent with the conventional reacting ideal-gas equation of state (the so-called ""Saha equation"") at low temperature and low densities, but (ii) that there is also a smooth matching with the FK virial expansion EOS up to the order of density ρ². Comparing our resulting combined equation of state (FK plus SLT) with currently popular equations of state, such as OPAL (developed at Livermore), the relative discrepancies of the relevant thermodynamic quantities are about 10⁻³ to 10⁻⁴. Since these differences are of the same order as the accuracy of helioseimic inversions for thermodynamic quantities, we have demonstrated that the FK equation of state extended by SLT will be a serious player in solar modeling. Further technical steps still have to be taken before the FK equation of state can be used in a fully-fledged helioseismic study. Our result demonstrates the feasibility of such an application, which will then tell whether the FK virial EOS is more accurate than the so-far best equation of state OPAL. With the exact nature of FK (devoid of several approximations used in OPAL), there is a promise of a successful outcome. However, at the moment, a final comparison is not possible because of the current state of the SLT EOS. Once it will be available for a multi-element plasmas, our product consisting of the SLT EOS for the solar exterior, and the FK virial EOS for the solar interior, will likely become the most accurate EOS for solar and stellar modeling.

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ON THE FEYNMAN PATH INTO THE SUN
by
Yung-Ching Liang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
August 2013
Copyright 2013 Yung-Ching Liang